Hinge theorem
In geometry, the hinge theorem (sometimes called the open mouth theorem) states that if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle.[1] This theorem is given as Proposition 24 in Book I of Euclid's Elements.
Scope and generalizations
The hinge theorem holds in Euclidean spaces and more generally in simply connected non-positively curved space forms.
It can be also extended from plane Euclidean geometry to higher dimension Euclidean spaces (e.g., to tetrahedra and more generally to simplices), as has been done for orthocentric tetrahedra (i.e., tetrahedra in which altitudes are concurrent)[2] and more generally for orthocentric simplices (i.e., simplices in which altitudes are concurrent).[3]
Converse
The converse of the hinge theorem is also true: If the two sides of one triangle are congruent to two sides of another triangle, and the third side of the first triangle is greater than the third side of the second triangle, then the included angle of the first triangle is larger than the included angle of the second triangle.
In some textbooks, the theorem and its converse are written as the SAS Inequality Theorem and the AAS Inequality Theorem respectively.
References
- ^ Moise, Edwin; Downs, Jr., Floyd (1991). Geometry. Addison-Wesley Publishing Company. p. 233. ISBN 0201253356.
- ^ Abu-Saymeh, Sadi; Mowaffaq Hajja; Mostafa Hayajneh (2012). "The open mouth theorem, or the scissors lemma, for orthocentric tetrahedra". Journal of Geometry. 103 (1): 1–16. doi:10.1007/s00022-012-0116-4.
- ^ Hajja, Mowaffaq; Mostafa Hayajneh (August 1, 2012). "The open mouth theorem in higher dimensions". Linear Algebra and Its Applications. 437 (3): 1057–1069. doi:10.1016/j.laa.2012.03.012.
- v
- t
- e
(timeline)
- Anaxagoras
- Anthemius
- Archytas
- Aristaeus the Elder
- Aristarchus
- Aristotle
- Apollonius
- Archimedes
- Autolycus
- Bion
- Bryson
- Callippus
- Carpus
- Chrysippus
- Cleomedes
- Conon
- Ctesibius
- Democritus
- Dicaearchus
- Diocles
- Diophantus
- Dinostratus
- Dionysodorus
- Domninus
- Eratosthenes
- Eudemus
- Euclid
- Eudoxus
- Eutocius
- Geminus
- Heliodorus
- Heron
- Hipparchus
- Hippasus
- Hippias
- Hippocrates
- Hypatia
- Hypsicles
- Isidore of Miletus
- Leon
- Marinus
- Menaechmus
- Menelaus
- Metrodorus
- Nicomachus
- Nicomedes
- Nicoteles
- Oenopides
- Pappus
- Perseus
- Philolaus
- Philon
- Philonides
- Plato
- Porphyry
- Posidonius
- Proclus
- Ptolemy
- Pythagoras
- Serenus
- Simplicius
- Sosigenes
- Sporus
- Thales
- Theaetetus
- Theano
- Theodorus
- Theodosius
- Theon of Alexandria
- Theon of Smyrna
- Thymaridas
- Xenocrates
- Zeno of Elea
- Zeno of Sidon
- Zenodorus
- Almagest
- Archimedes Palimpsest
- Arithmetica
- Conics (Apollonius)
- Catoptrics
- Data (Euclid)
- Elements (Euclid)
- Measurement of a Circle
- On Conoids and Spheroids
- On the Sizes and Distances (Aristarchus)
- On Sizes and Distances (Hipparchus)
- On the Moving Sphere (Autolycus)
- Optics (Euclid)
- On Spirals
- On the Sphere and Cylinder
- Ostomachion
- Planisphaerium
- Spherics (Theodosius)
- Spherics (Menelaus)
- The Quadrature of the Parabola
- The Sand Reckoner
and definitions
- Angle
- Axiomatic system
- Chord
- Circles of Apollonius
- Circumscribed circle
- Commensurability
- Diophantine equation
- Doctrine of proportionality
- Euclidean geometry
- Golden ratio
- Greek numerals
- Incircle and excircles of a triangle
- Method of exhaustion
- Parallel postulate
- Platonic solid
- Lune of Hippocrates
- Quadratrix of Hippias
- Regular polygon
- Straightedge and compass construction
- Triangle center
History of | |
---|---|
Other cultures |